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For 3x3 real matrices A and B, identify which of the following statements are correct.

1. AB is skew-symmetric whenever A is symmetric and B is skew-symmetric
2. (adj A)^T = adj(A^T) for every invertible matrix A
3. AB + BA is symmetric for all symmetric matrices A and B
4. (adj A)^(-1) = adj(A^(-1)) for every invertible matrix A

Correct answer: AB + BA is symmetric for all symmetric matrices A and B

Solution

Statements B, C, and D are all true. (AB+BA)^T = B^T A^T + A^T B^T = BA + AB = AB + BA, so C holds. B and D follow from the identity adj(A) = det(A)*A^(-1). Statement A is false in general since (AB)^T = -BA not -AB.

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